%I #20 Oct 29 2019 09:06:48
%S 1,-10,1,110,-21,1,-1320,362,-33,1,17160,-6026,791,-46,1,-240240,
%T 101524,-17100,1435,-60,1,3603600,-1763100,358024,-38625,2335,-75,1,
%U -57657600,31813200,-7491484,976024,-75985,3535,-91,1,980179200,-598482000,159168428,-24083892,2267769,-136080,5082,-108,1
%N Generalized Stirling number triangle of first kind.
%C a(n,m)= ^10P_n^m in the notation of the given reference with a(0,0) := 1. The monic row polynomials s(n,x) := sum(a(n,m)*x^m,m=0..n) which are s(n,x)= product(x-(10+k),k=0..n-1), n >= 1 and s(0,x)=1 satisfy s(n,x+y) = sum(binomial(n,k)*s(k,x)*S1(n-k,y),k=0..n), with the Stirling1 polynomials S1(n,x)=sum(A008275(n,m)*x^m, m=1..n) and S1(0,x)=1. In the umbral calculus (see the S. Roman reference given in A048854) the s(n,x) polynomials are called Sheffer for (exp(10*t),exp(t)-1).
%H Reinhard Zumkeller, <a href="/A051523/b051523.txt">Rows n = 0..125 of triangle, flattened</a>
%H D. S. Mitrinovic, M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
%F a(n, m)= a(n-1, m-1) - (n+9)*a(n-1, m), n >= m >= 0; a(n, m) := 0, n<m; a(n, -1) := 0, a(0, 0)=1.
%F E.g.f. for m-th column of signed triangle: ((log(1+x))^m)/(m!*(1+x)^10).
%F Triangle (signed) = [ -10, -1, -11, -2, -12, -3, -13, -14, -4, ...] DELTA A000035; triangle (unsigned) = [10, 1, 11, 2, 12, 3, 13, 4, 14, 5, 15, ...] DELTA A000035; where DELTA is Deléham's operator defined in A084938.
%F If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then T(n,i) = f(n,i,10), for n=1,2,...;i=0...n. - _Milan Janjic_, Dec 21 2008
%e {1}; {-10,1}; {110,-21,1}; {-1320,362,-331}; ... s(2,x)= 110-21*x+x^2; S1(2,x)= -x+x^2 (Stirling1).
%t a[n_, m_] := Pochhammer[m + 1, n - m] SeriesCoefficient[Log[1 + x]^m/(1 + x)^10, {x, 0, n}];
%t Table[a[n, m], {n, 0, 8}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 29 2019 *)
%o (Haskell)
%o a051523 n k = a051523_tabl !! n !! k
%o a051523_row n = a051523_tabl !! n
%o a051523_tabl = map fst $ iterate (\(row, i) ->
%o (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 10)
%o -- _Reinhard Zumkeller_, Mar 12 2014
%Y The first (m=0) unsigned column sequence is A049398. Row sums (signed triangle): A049389(n)*(-1)^n. Row sums (unsigned triangle): A051431(n).
%Y See also A049444 A049458 A049459 A049460 A051338 A051339 A051379 A051380
%Y Cf. A000035 A084938.
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_